Pauli Spin Blockade and the Ultrasmall Magnetic Field Effect

Jeroen Danon,1 Xuhui Wang,2 and Aurelien Manchon2

1Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark2King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division,

Thuwal 23955-6900, Saudi Arabia(Received 23 March 2013; published 6 August 2013)

Based on the spin-blockade model for organic magnetoresistance, we present an analytic expression

for the polaron-bipolaron transition rate, taking into account the effective nuclear fields on the two sites.

We reveal the physics behind the qualitatively different magnetoconductance line shapes observed in

experiment, as well as the ultrasmall magnetic field effect (USFE). Since our findings agree in detail with

recent experiments, they also indirectly provide support for the spin-blockade interpretation of organic

magnetoresistance. In addition, we predict the existence of a similar USFE in semiconductor double

quantum dots tuned to the spin-blockade regime.

DOI: 10.1103/PhysRevLett.111.066802 PACS numbers: 73.50.�h, 71.38.Mx, 73.43.Qt

The discovery some ten years ago of spin injection inorganic semiconductors [1] and a giant magnetoresistance inorganic spin valves [2,3] triggered the birth of the thrivingfield of organic spintronics [4], which offers interesting newphysics and the potential of industrial applications. An excit-ing phenomenon in this field is a large (up to 20%) magne-toresistance observed in different organic materials [5–7],usually at small magnetic fields (1–10 mT) but sometimes atlarger fields (10–100 mT) and persisting up to room tem-perature. Since its discovery in 2004, different explanationsfor this organic magnetoresistance (OMAR) have beenproposed: For bipolar devices, it was suggested that spin-dependent electron-hole recombination and dissociationrates could be responsible [8,9], whereas a model based onnuclear-field-mediated bipolaron formation could explainOMAR in both bipolar and unipolar devices [10–12].

More recently, an organic magnetoresistive effect on aneven smaller field scale (0.1–1 mT) has been observed inunipolar as well as bipolar devices [13]. This ultrasmallmagnetic field effect (USFE) ismanifested by a sign reversalof the magnetoconductance (MC) at very small fields (scal-ingwith thewidth of theMCcurve), which creates two smallpeaks (dips) around zero field for devices with a negative(positive) MC [13]. An explanation was suggested in termsof enhanced singlet-triplet mixing close to the crossings ofthe hyperfine sublevels of pairs of charge carriers (polarons)coupled to single nuclear spins [14]. This explanation is stillunder debate, mainly because it is expected that a singlepolaron in reality couples to many nuclear spins [15–17].Others attribute the USFE to the interplay of the polaronZeeman and exchange splittings [18], which could be rele-vant for materials with strong coherent intersite coupling.

Here, we study the OMAR line shape as it naturallyemerges from the spin-blockade model of Ref. [10]. Wepresent an analytic expression for the charge currentthrough a polaron-bipolaron link for a given realizationof the nuclear fields. Our results reproduce the USFE and

the different linewidths as observed in experiment and innumerical calculations based on the same semiclassicalapproach [19], and from our analytic insight we can iden-tify the underlying physical mechanisms. We thus not onlyreveal the physical origin of the USFE but thereby alsoprovide support for the spin-blockade model capturing theunderlying physics of OMAR.We note that many interesting aspects of spin-blockade

physics have already been investigated in the seeminglyforeign field of spin qubits hosted in semiconductor quan-tum dots [20,21], where spin blockade is commonly usedas a tool for single-qubit readout [22,23]. Indeed, thephysics of the polaron spin-blockade model for OMAR isvery similar to that governing the electron transportthrough a double quantum dot in the spin-blockade regime[21]. Our investigation thus builds on the theoretical frame-work of Ref. [21], and our explanation of the USFE relieson a subtlety which was not addressed in Ref. [21]. Wetherefore also provide a refined understanding of quantumdot spin-blockade physics and claim that the USFE alsomust be present in electron transport through double quan-tum dots. Indeed, a close inspection of the experimentaldata presented in Ref. [21] also seems to reveal a faintUSFE, which always has been ignored. In fact, due to itstunability, we expect that a double quantum dot might bethe best system to experimentally explore the USFE inmore detail.Let us first briefly review the bipolaron model for

OMAR presented in Ref. [10]. Electric current flowsthrough the organic material as polarons hop betweendifferent localized molecular sites. Typically, the sitesparticipating in transport do not form a regular lattice,and all have a random energy offset with a distributionwidth � of 0.1–0.2 eV [10]. Sites with a relatively largenegative energy offset are likely to trap a polaron for a longtime, but since the on-site polaron-polaron repulsion istypically of the same order of magnitude as �, such

PRL 111, 066802 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

9 AUGUST 2013

0031-9007=13=111(6)=066802(5) 066802-1 � 2013 American Physical Society

occupied sites can often still take part in transport bytemporarily hosting a pair of polarons, i.e., a bipolaron.

Because of a relatively large orbital level spacing,most energetically accessible bipolaron states are spinsinglets. This makes the polaron-bipolaron transition spinselective, ultimately leading to OMAR. The mechanismcan be understood from Fig. 1, where we focus on a singlepolaron-bipolaron transition. We assume that the spins ofthe two encountering polarons are random, and for sim-plicity we describe the problem in the basis of spin eigen-states quantized along the direction of the local magneticfields BL;R. Two possible initial spin states are depicted:

(i) the left spin antiparallel and the right spin parallel tothe local field and (ii) both spins parallel. In the absence ofan external field, the magnetic fields at the two sites are thelocal random effective nuclear fields [Fig. 1(a)]. Generally,all initial states can then transition to a spin-singlet bipo-laron and current runs through the system. If, on the otherhand, a magnetic field much larger than the typical nuclearfields is applied, then BL and BR are (almost) parallel[Fig. 1(b)]. In this case, situation (ii) is a spin triplet, outof which a bipolaron cannot be formed: the current isblocked. Of course, in experiment, there are many possiblepaths for charge carriers through the material and not all ofthem contain bipolaron sites. The visibility of all effects ofspin blockade will thus be reduced, but the characteristicfeatures survive [10].

In this work, we will focus on the physics of a singlepolaron-bipolaron transition and its MC line shape. Todescribe the transition, we use five states: the four possibleinitial spin states of the polaron pair (both sites hosting onepolaron)—one spin-singlet state jSi and three spin-tripletstates jT0i and jT�i—and the spin-singlet bipolaron statejSbi. The Hamiltonian we use to describe the coherentdynamics of these states reads [21]

H ¼

Bzs B�

s 0 �B�a 0

Bþs 0 B�

s Bza 0

0 Bþs �Bz

s Bþa 0

�Bþa Bz

a B�a 0 t

0 0 0 t ��

0BBBBBBBB@

1CCCCCCCCA; (1)

written in the basis fjTþi; jT0i; jT�i; jSi; jSbig. ThisHamiltonian includes a coupling energy t between thetwo singlets (which enables polaron hopping) and therelative energy offset (detuning) � of the bipolaron state,typically �� �. The effect of the local magnetic fieldsBL;R is expressed in terms of the sum and difference

fields Bs ¼ ð1=2ÞðBL þBRÞ and Ba ¼ ð1=2ÞðBL �BRÞ,and we use the notation B�

sðaÞ ¼ ð1= ffiffiffi2

p ÞðBxsðaÞ � iBy

sðaÞÞ.Note that we have set g�B ¼ 1 for convenience.As pointed out in Ref. [21], we can deduce already from

Eq. (1) that there exist in the space of (BL, BR) so-called‘‘stopping points’’ where the current is blocked. To see this,we take the spin quantization axis to point along Bs, whichamounts to setting B�

s ! 0 in Eq. (1). Then, we find thatcurrent vanishes when Ba k Bs or Ba ? Bs, since at thesepoints one or more of the triplet states are not coupled tojSi. The sum and difference fields Bs;a both contain a

contribution from the effective nuclear fields KL;R on the

two sites, whereas the external field Bext only adds to thesum field: Bs ¼ Ks þ Bextz and Ba ¼ Ka. For a givenrandom realization of KL;R, one can thus always find

a field Bext for which Ba ? Bs, and a sweep of Bext for afixed KL;R will always exhibit a stopping point where the

current vanishes. The position of this stopping point isdetermined by the relative orientations of Ks and Ka andis thus random. In an experiment, one usually sweeps Bext

so slowly that at each measurement, many configurations ofthe fields KL;R are probed. As a result, the stopping points

are averaged out and one finds a smooth MC curve [21].However, this is not the full story. A subtlety, not dis-

cussed in Ref. [21], is that there exists one more stoppingpoint [24]: When Bs ¼ 0, the triplet subspace in theHamiltonian is degenerate and Eq. (1) can be equivalentlywritten in terms of one coupled triplet state

jTmi ¼ �B�a jTþi þ Bz

ajT0i þ Bþa jT�i

jBajand two orthogonal triplet states jT1i and jT2i which have

hT1;2jHjSi ¼ 0 and are thus blocked. Why would we

bother? We argued above that stopping points occur atrandom positions, leaving no trace after averaging overKL;R. This new stopping point, however, is fundamentally

different from the ones discussed above: It suppressescurrent close to where Bext¼�Ks, which is always in thevicinity of Bext ¼ 0. It is therefore possible that after aver-aging over KL;R, this new stopping point leaves a trace in

the MC curve: a small dip around zero field, like the USFE.Let us now explicitly calculate the current as governed

by this polaron-bipolaron transition. To describe chargetransport, we write a time-evolution equation for the5� 5 density matrix of the system. To the coherent evo-

lution dictated by H, we add incoherent rates describingdissociation of the bipolaron to the environment and hop-ping of a new polaron onto the empty left site. This yields(where we have set @ to 1)

(a) (b)

FIG. 1 (color online). When a site already contains a polaron(the right site in the pictures), charge transport through this siterelies on the formation of a bipolaron (thick blue arrows). Thisbipolaron must be a spin singlet, which leads to spin blockade.

PRL 111, 066802 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

9 AUGUST 2013

066802-2

@�

@t¼ �i½H; �� � �

2fPb; �g þ �

4�b;bð1� PbÞ1; (2)

where � is the rate of bipolaron dissociation to the environ-

ment, Pb ¼ jSbihSbj is the projection operator onto the

bipolaron state, and 1 is the identity matrix. In writingso, we assumed for simplicity that the refilling of the leftsite takes place immediately after the dissociation of thebipolaron. If this is not the case, the prefactor for thecurrent changes but the MC characteristics stay the same.

We add the normalization condition Tr½�� ¼ 1 to the set

of equations and then solve @t�ðeqÞ ¼ 0 to find the sta-

tionary density matrix. The charge current is then given by

I ¼ e��ðeqÞb;b and can be found explicitly. We assume for

convenience that � � t, Bs, Ba, � is the largest energyscale in the problem [12], and then find

I ¼ e�s

4x2sin2�

x4 þ ax2 þ 1; (3)

in terms of x � Bs=Ba. Here, �s � t2=� is the singlet-singlet hopping rate from the left to the right site and �is the angle between Bs and Ba [25]. We also used

a ¼ �2s

B2a

�3þ 1

cos2�

�� 2 cos2�: (4)

We see that all stopping points predicted above are indeedreflected in Eq. (3): At Bs ¼ 0, we have x¼0, which yieldsI ¼ 0, and Ba k Bs or Ba ? Bs corresponds to � ¼ 0, �or � ¼ �=2, respectively, both also giving I ¼ 0.

Equation (3) is the most important analytic result of ourwork. It gives the current for one single realization of KL,KR, and Bext. The MC measured in experiment is found byaveraging Eq. (3) over the random nuclear fields. In con-trast to the analytic results presented in Ref. [21], our resultis valid for arbitrary �s and not only for limiting cases. Oneword of caution is required here concerning the interpre-tation of Eq. (3): If one wants to plot IðBextÞ for a singlerealization of KL;R, one should not only use Bs ¼ jKs þBextzj in Eq. (3) but also implement the dependence of �on Bext implied by cos� ¼ ðBs � BaÞ=BsBa.

Let us now investigate Eq. (3) and see what we can inferabout the line shape of the predicted MC curve. We alwayshave a >�2, which ensures that I 0 everywhere. Thecurrent vanishes for x ¼ 0 or x ! 1, and in the range x 2½0;1�, we have a single maximum at x ¼ 1 where thecurrent is Imax ¼ 4e�ssin

2�=ðaþ 2Þ. In Fig. 2(a), weplot the expression given in Eq. (3) for different a. TheFWHMs w� of the dip around x ¼ 0 and wþ of the overallpeak structure (as indicated in the plot for a ¼ 4) are found

to be w2� ¼ 2þ ð1=2Þa� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ 2aþ ð1=4Þa2p

.We see from Eq. (4) that an important parameter is

�s=K, the ratio of the intersite hopping rate and the typicalmagnitude of the nuclear fields K, typically �0:1 �eV[15]. We will thus now investigate the cases of small andlarge �s=K.

In the limit of �s=K 1, we can write

I � e�s

4x2sin2�

x4 � 2x2 cos2�þ 1¼ �sðnL � nRÞ2; (5)

where we used the unit vectors nL;R ¼ BL;R=BL;R. As it

should, this result coincides with that of Ref. [21] in thesame limit: There are no intersite exchange effects, andthe situation is exactly like the picture of Fig. 1 where thecurrent only depends on the relative orientations of BL andBR. As was shown in Ref. [21], Eq. (5) can be averagedanalytically over random KL;R taken from a normal distri-

bution, yielding a MC curve with a flat peak at Bext ¼ 0, amaximum of hIimax � e�s, and a linewidth of�K. Indeed,for all a 2 ½�2; 2�, we find that wþ � 1, so for any � thecurrent is suppressed when x * 1. In Fig. 2(b) (blue trace),we plot the resulting MC line shape, where we definedMCðBextÞ ¼ ½IðBextÞ � Ið0Þ�=Ið0Þ.In the opposite limit of �s=K � 1, we have a �

ð�s=BaÞ2ð3þ cos�2�Þ � 1. We can already see from theproperties of Eq. (3) that in this case, hIimax � eK2=�s, and

that wþ � a1=2 � �s=K implies a MC linewidth of ��s.Indeed, �s sets the level broadening of jSi, and as long asBs & �s, generally all three triplet states can efficientlytransition to jSi with the coupling provided by Ba. Thewidth of the dip around the stopping point at x ¼ 0 is

w� � a�1=2 � K=�s in terms of x, or �K2=�s in termsof Bs. This energy scale can also be understood: If Bs ¼ 0,the decay rate of jTmi is �t � K2=�s, which is the onlyenergy relevant in the triplet subspace. When Bs * �t, the

(a) (b)

(c)

FIG. 2 (color online). (a) The current given by Eq. (3) as afunction of x ¼ Bs=Ba for fixed Ba and �, evaluated for differ-ent parameters a. (b),(c) The averaged MC. (b) The bluetrace represents �s=K 1. The field Bext is plotted in units ofK ¼ hK2

L;Ri1=2. The peak of this curve is flat [21]. The green

trace represents �s=K ¼ 3=2. (c) �s=K ¼ 50. Now, Bext isplotted in units of �s. Inset: The range where Bext � K.

PRL 111, 066802 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

9 AUGUST 2013

066802-3

decay of the other two triplet states becomes comparable to�t and the blockade is lifted.

In this limit of large �s=K, the current cannot be aver-aged analytically over the nuclear fields, and one has toevaluate the integrals over the distribution ofKL;R numeri-

cally. Figure 2(c) shows a plot of the MC integrated overnormal distributions for all six components ofKL andKR.

For all components, we used a standard deviation of K=ffiffiffi3

pand we have set �s=K ¼ 50. The resulting line shape isLorentzian since it is determined by the level broadening ofjSi. Close to zero field, where Bext � K, we find a very faintUSFE, as shown in the inset. When we set �s=K evenlarger, we find that the visibility of this USFE is suppressedfurther, ultimately reaching zero.

We can understand this USFE from the expression forthe current given in Eq. (3). For a given realization ofKL;R,

the current trace IðBextÞ ‘‘misses’’ the zero-field stopping

point by K?s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKx

s Þ2 þ ðKys Þ2

p, as illustrated in Fig. 3.

Some realizations have K?s Ba so that the current trace

has a single maximum [see Fig. 2(a)]. Other realizationshave K?

s < Ba, and the current exhibits a dip at smallfields, the position of the dip at Bext ¼ �Kz

s . For large�s=K, the dip around the zero-field stopping point becomesnarrow, of the order of �K2=�s K, and only the veryfew curves of IðBextÞ with K?

s & K2=�s have an appre-ciable dip. This can still produce a faint dip in the averagedcurrent. However, the position of each single-realizationnarrow dip is at Bext ¼ �Kz

s , so averaging over Kzs makes

the averaged dip even less pronounced and results in a dipwidth of �K.

The regime to look for a pronounced USFE is thus atintermediate �s=K � 1. In Fig. 2(b) (green trace), we plotthe averaged MC for �s=K ¼ 3=2 and we see indeed astrong USFE, its visibility being �5%. This regime isoptimal for the USFE since here the width of the zero-fielddip is still �K, but the symmetric situation where thecurrent only depends on the angle between nL and nR issignificantly perturbed. In other words, at �s=K ! 0, theoverall MC linewidth is minimal and �K. The two USFE‘‘bumps’’ are still there but are split by the same energyscale �K and thus appear just left and right of the top of

the MC curve. In the limit of �s=K ¼ 0, the bumps and theunderlying MC curve have exactly compatibly shaped lineshapes, together resulting in the characteristic flat peak.If one moves away from �s=K ¼ 0, the underlying MCline shape becomes broader, which makes the USFEbumps more visible. However, as soon as �s=K becomestoo large, one enters the regime discussed above, wherethe USFE disappears again. The optimal regime is thus at�s=K � 1, in agreement with the results presented inFigs. 2(b) and 2(c) as well as with previously obtainednumerical results [12].To summarize, we studied the two-site spin-blockade

model for OMAR and derived an analytic expression forthe polaron-bipolaron transition rate, which provides anexplanation of the USFE in terms of a persistent spinblockade at the special point where the average effectivefield vanishes Bs ¼ 0. Our work also deepens the under-standing of spin-blockade physics in semiconductor quan-tum dots, and we predict the existence of an USFE in thesesystems as well.We acknowledge helpful feedback from M. S. Rudner

and P.A. Bobbert.

[1] V. Dediu, M. Murgia, F. Matacotta, C. Taliani, and S.Barbanera, Solid State Commun. 122, 181 (2002).

[2] J. Kalinowski, M. Cocchi, D. Virgili, P. Di Marco, andV. Fattori, Chem. Phys. Lett. 380, 710 (2003).

[3] Z. H. Xiong, D. Wu, Z. V. Vardeny, and J. Shi, Nature(London) 427, 821 (2004).

[4] V. A. Dediu, L. E. Hueso, I. Bergenti, and C. Taliani,Nat. Mater. 8, 707 (2009).

[5] T. L. Francis, O. Mermer, G. Veeraraghavan, and M.Wohlgenannt, New J. Phys. 6, 185 (2004).

[6] O. Mermer, G. Veeraraghavan, T. L. Francis, Y. Sheng,D. T. Nguyen, M. Wohlgenannt, A. Kohler, M.K. Al-Suti,and M. S. Khan, Phys. Rev. B 72, 205202 (2005).

[7] F. L. Bloom, W. Wagemans, M. Kemerink, and B.Koopmans, Phys. Rev. Lett. 99, 257201 (2007).

[8] V. Prigodin, J. Bergeson, D. Lincoln, and A. Epstein,Synth. Met. 156, 757 (2006).

[9] P. Desai, P. Shakya, T. Kreouzis, and W. P. Gillin, Phys.Rev. B 76, 235202 (2007).

[10] P. A. Bobbert, T. D. Nguyen, F.W.A. van Oost, B.Koopmans, and M. Wohlgenannt, Phys. Rev. Lett. 99,216801 (2007).

[11] W.Wagemans, F. L.Bloom, P.A.Bobbert,M.Wohlgenannt,and B. Koopmans, J. Appl. Phys. 103, 07F303 (2008).

[12] A. J. Schellekens, W. Wagemans, S. P. Kersten, P. A.Bobbert, andB.Koopmans, Phys.Rev.B 84, 075204 (2011).

[13] T.D. Nguyen, B. R. Gautam, E. Ehrenfreund, and Z.V.Vardeny, Phys. Rev. Lett. 105, 166804 (2010).

[14] T.D. Nguyen, G. Hukic-Markosian, F. Wang, L. Wojcik,X.-G. Li, E. Ehrenfreund, and Z.V. Vardeny, Nat. Mater. 9,345 (2010).

[15] D. R. McCamey, K. J. van Schooten, W. J. Baker, S.-Y.Lee, S.-Y. Paik, J.M. Lupton, and C. Boehme, Phys. Rev.Lett. 104, 017601 (2010).

FIG. 3 (color online). When Bext is swept for a given realiza-tion of KL;R, the field Bs (blue arrow) follows a trace like the

dashed blue line. We indicated with green arrows Ks as well asK?

s , which equals the minimum value of Bs.

PRL 111, 066802 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

9 AUGUST 2013

066802-4

[16] P. A. Bobbert, Nat. Mater. 9, 288 (2010).[17] K. Schulten and P.G. Wolynes, J. Chem. Phys. 68, 3292

(1978).[18] N. J. Harmon and M. E. Flatte, Phys. Rev. B 85, 245213

(2012).[19] S. P. Kersten, A. J. Schellekens, B. Koopmans, and P. A.

Bobbert, Phys. Rev. Lett. 106, 197402 (2011).[20] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science

297, 1313 (2002).[21] O. N. Jouravlev and Y.V. Nazarov, Phys. Rev. Lett. 96,

176804 (2006).[22] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,

K. C. Nowack, T. Meunier, L. P. Kouwenhoven, andL.M.K. Vandersypen, Nature (London) 442, 766 (2006).

[23] D. J. Reilly, J.M. Taylor, J. R. Petta, C.M. Marcus,M. P. Hanson, and A. C. Gossard, Science 321, 817(2008).

[24] One could argue that Ba ¼ 0 is also a stopping point, butit is not of any interest since it occurs independently fromBext and therefore leaves no traces in the MC curve. In thatsense, it is also not really a stopping point. Besides, it iscaptured by the model in Ref. [21] since it is equivalent tohaving Ba k Bs and Ba ? Bs.

[25] If one would relax the assumption � � �, one finds thesame expression but with �s ! t2=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ 4�2

pand an extra

prefactor �=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ 4�2

p. A finite detuning � thus merely

leads to a suppression of the current as well as a smallereffective hopping rate.

PRL 111, 066802 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

9 AUGUST 2013

066802-5